3.354 \(\int \frac{x^4 (A+B x)}{\sqrt{a+c x^2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{5/2}}+\frac{a \sqrt{a+c x^2} (64 a B-45 A c x)}{120 c^3}+\frac{A x^3 \sqrt{a+c x^2}}{4 c}-\frac{4 a B x^2 \sqrt{a+c x^2}}{15 c^2}+\frac{B x^4 \sqrt{a+c x^2}}{5 c} \]

[Out]

(-4*a*B*x^2*Sqrt[a + c*x^2])/(15*c^2) + (A*x^3*Sqrt[a + c*x^2])/(4*c) + (B*x^4*S
qrt[a + c*x^2])/(5*c) + (a*(64*a*B - 45*A*c*x)*Sqrt[a + c*x^2])/(120*c^3) + (3*a
^2*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*c^(5/2))

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Rubi [A]  time = 0.359648, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 a^2 A \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{5/2}}+\frac{a \sqrt{a+c x^2} (64 a B-45 A c x)}{120 c^3}+\frac{A x^3 \sqrt{a+c x^2}}{4 c}-\frac{4 a B x^2 \sqrt{a+c x^2}}{15 c^2}+\frac{B x^4 \sqrt{a+c x^2}}{5 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(A + B*x))/Sqrt[a + c*x^2],x]

[Out]

(-4*a*B*x^2*Sqrt[a + c*x^2])/(15*c^2) + (A*x^3*Sqrt[a + c*x^2])/(4*c) + (B*x^4*S
qrt[a + c*x^2])/(5*c) + (a*(64*a*B - 45*A*c*x)*Sqrt[a + c*x^2])/(120*c^3) + (3*a
^2*A*ArcTanh[(Sqrt[c]*x)/Sqrt[a + c*x^2]])/(8*c^(5/2))

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Rubi in Sympy [A]  time = 35.0483, size = 121, normalized size = 0.94 \[ \frac{3 A a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{a + c x^{2}}} \right )}}{8 c^{\frac{5}{2}}} + \frac{A x^{3} \sqrt{a + c x^{2}}}{4 c} - \frac{4 B a x^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{B x^{4} \sqrt{a + c x^{2}}}{5 c} + \frac{a \sqrt{a + c x^{2}} \left (- 45 A c x + 64 B a\right )}{120 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(B*x+A)/(c*x**2+a)**(1/2),x)

[Out]

3*A*a**2*atanh(sqrt(c)*x/sqrt(a + c*x**2))/(8*c**(5/2)) + A*x**3*sqrt(a + c*x**2
)/(4*c) - 4*B*a*x**2*sqrt(a + c*x**2)/(15*c**2) + B*x**4*sqrt(a + c*x**2)/(5*c)
+ a*sqrt(a + c*x**2)*(-45*A*c*x + 64*B*a)/(120*c**3)

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Mathematica [A]  time = 0.105656, size = 89, normalized size = 0.69 \[ \frac{\sqrt{a+c x^2} \left (64 a^2 B-a c x (45 A+32 B x)+6 c^2 x^3 (5 A+4 B x)\right )+45 a^2 A \sqrt{c} \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{120 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(A + B*x))/Sqrt[a + c*x^2],x]

[Out]

(Sqrt[a + c*x^2]*(64*a^2*B + 6*c^2*x^3*(5*A + 4*B*x) - a*c*x*(45*A + 32*B*x)) +
45*a^2*A*Sqrt[c]*Log[c*x + Sqrt[c]*Sqrt[a + c*x^2]])/(120*c^3)

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Maple [A]  time = 0.011, size = 117, normalized size = 0.9 \[{\frac{A{x}^{3}}{4\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,aAx}{8\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,A{a}^{2}}{8}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{x}^{4}B}{5\,c}\sqrt{c{x}^{2}+a}}-{\frac{4\,aB{x}^{2}}{15\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{8\,{a}^{2}B}{15\,{c}^{3}}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(B*x+A)/(c*x^2+a)^(1/2),x)

[Out]

1/4*A*x^3*(c*x^2+a)^(1/2)/c-3/8*A*a/c^2*x*(c*x^2+a)^(1/2)+3/8*A*a^2/c^(5/2)*ln(c
^(1/2)*x+(c*x^2+a)^(1/2))+1/5*B*x^4*(c*x^2+a)^(1/2)/c-4/15*a*B*x^2*(c*x^2+a)^(1/
2)/c^2+8/15*B*a^2/c^3*(c*x^2+a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^4/sqrt(c*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.354413, size = 1, normalized size = 0.01 \[ \left [\frac{45 \, A a^{2} c \log \left (-2 \, \sqrt{c x^{2} + a} c x -{\left (2 \, c x^{2} + a\right )} \sqrt{c}\right ) + 2 \,{\left (24 \, B c^{2} x^{4} + 30 \, A c^{2} x^{3} - 32 \, B a c x^{2} - 45 \, A a c x + 64 \, B a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{c}}{240 \, c^{\frac{7}{2}}}, \frac{45 \, A a^{2} c \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (24 \, B c^{2} x^{4} + 30 \, A c^{2} x^{3} - 32 \, B a c x^{2} - 45 \, A a c x + 64 \, B a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-c}}{120 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^4/sqrt(c*x^2 + a),x, algorithm="fricas")

[Out]

[1/240*(45*A*a^2*c*log(-2*sqrt(c*x^2 + a)*c*x - (2*c*x^2 + a)*sqrt(c)) + 2*(24*B
*c^2*x^4 + 30*A*c^2*x^3 - 32*B*a*c*x^2 - 45*A*a*c*x + 64*B*a^2)*sqrt(c*x^2 + a)*
sqrt(c))/c^(7/2), 1/120*(45*A*a^2*c*arctan(sqrt(-c)*x/sqrt(c*x^2 + a)) + (24*B*c
^2*x^4 + 30*A*c^2*x^3 - 32*B*a*c*x^2 - 45*A*a*c*x + 64*B*a^2)*sqrt(c*x^2 + a)*sq
rt(-c))/(sqrt(-c)*c^3)]

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Sympy [A]  time = 16.6752, size = 173, normalized size = 1.34 \[ - \frac{3 A a^{\frac{3}{2}} x}{8 c^{2} \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{A \sqrt{a} x^{3}}{8 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 c^{\frac{5}{2}}} + \frac{A x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + B \left (\begin{cases} \frac{8 a^{2} \sqrt{a + c x^{2}}}{15 c^{3}} - \frac{4 a x^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{x^{4} \sqrt{a + c x^{2}}}{5 c} & \text{for}\: c \neq 0 \\\frac{x^{6}}{6 \sqrt{a}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(B*x+A)/(c*x**2+a)**(1/2),x)

[Out]

-3*A*a**(3/2)*x/(8*c**2*sqrt(1 + c*x**2/a)) - A*sqrt(a)*x**3/(8*c*sqrt(1 + c*x**
2/a)) + 3*A*a**2*asinh(sqrt(c)*x/sqrt(a))/(8*c**(5/2)) + A*x**5/(4*sqrt(a)*sqrt(
1 + c*x**2/a)) + B*Piecewise((8*a**2*sqrt(a + c*x**2)/(15*c**3) - 4*a*x**2*sqrt(
a + c*x**2)/(15*c**2) + x**4*sqrt(a + c*x**2)/(5*c), Ne(c, 0)), (x**6/(6*sqrt(a)
), True))

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GIAC/XCAS [A]  time = 0.277225, size = 117, normalized size = 0.91 \[ \frac{1}{120} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \,{\left (\frac{4 \, B x}{c} + \frac{5 \, A}{c}\right )} x - \frac{16 \, B a}{c^{2}}\right )} x - \frac{45 \, A a}{c^{2}}\right )} x + \frac{64 \, B a^{2}}{c^{3}}\right )} - \frac{3 \, A a^{2}{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*x^4/sqrt(c*x^2 + a),x, algorithm="giac")

[Out]

1/120*sqrt(c*x^2 + a)*((2*(3*(4*B*x/c + 5*A/c)*x - 16*B*a/c^2)*x - 45*A*a/c^2)*x
+ 64*B*a^2/c^3) - 3/8*A*a^2*ln(abs(-sqrt(c)*x + sqrt(c*x^2 + a)))/c^(5/2)